arXiv subjects

Nicolai Jork

Publications and source records attributed to Nicolai Jork.

12 recordsLinked to original sources

arXiv2026

On quantitative sufficient second-order optimality conditions for elliptic optimal control problems

In this paper, a quantitative condition for optimality for distributed optimal control problems with box-constraints that are subject to a semilinear elliptic equation is considered. An important property of the investigated optimal control problems is the absence of a Tikhonov regularization. It is well known that at a given control, the second variation is a quadratic form in the linearized state, and its curvature coefficient function may vanish or change sign. We present a quantitative condition that implies coercivity with respect to the $L^2$-norm of the linearized-states. As a consequence, the stability of the second-order condition under perturbations of the states and the tracking data is shown.

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arXiv2026

Differentiability of the value function in control-constrained parabolic problems

Along the optimal trajectory of an optimal control problem constrained by a semilinear parabolic partial differential equation, we prove the differentiability of the value function with respect to the initial condition and, under additional assumptions on the solution of the state equation, the differentiability of the value function with respect to the time variable. In our proof, we rely on local growth assumptions commonly associated with the study of second-order sufficient conditions. These assumptions are generally applicable to a wide range of problems, including, for instance, certain tracking-type problems. Finally, we discuss the differentiability of the value function in a neighborhood of the optimal trajectory when a growth condition for optimal controls is used.

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arXiv2026

Error estimates for an unregularized optimal control problem for the stationary Navier-Stokes equations

We consider an unregularized optimal control problem subject to the steady-state Navier-Stokes equations. We derive the existence of optimal solutions and prove first- and second-order optimality conditions. To approximate solutions to the optimal control problem, we consider the variational discretization scheme. We analyze convergence properties of the discretization and prove a priori error estimates for locally optimal controls that are nonsingular and which satisfy a growth condition which implies a bang-bang structure. We also propose a residual-type a posteriori error estimator that accounts for the discretization of the state and adjoint equations, and prove suitable reliability properties for such an error estimator.

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arXiv2026

Second-Order Conditions for Infinite-Horizon Semilinear Parabolic Control Problems without Tikhonov Regularization

We consider semilinear parabolic optimal control problems subject to Neumann boundary conditions, control constraints, and an infinite time horizon. The control constraints are pointwise in time, but they can be pointwise or integral in the space variable. Crucially, the optimal control problem does not include a Tikhonov regularization in the cost functional, which provides a major difficulty in the extension of the classical finite-horizon theory to infinite-horizon optimal control problems. As a consequence of our findings, we establish a sufficient second-order optimality condition and prove that local optimal states of the finite-horizon problems approximate local optimal states to the infinite-horizon problem as the horizon tends to infinity.

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arXiv2025

Adaptive finite element method for an unregularized semilinear optimal control problem

We devise an a posteriori error estimator for an affine optimal control problem subject to a semilinear elliptic PDE and control constraints. To approximate the problem, we consider a semidiscrete scheme based on the variational discretization approach. For this solution technique, we design an a posteriori error estimator that accounts for the discretization of the state and adjoint equations, and prove, under suitable local growth conditions of optimal controls, reliability and efficiency properties of such error estimator. A simple adaptive strategy based on the devised estimator is designed and its performance is illustrated with numerical examples.

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arXiv2024

Finite element error analysis of affine optimal control problems

This paper is concerned with error estimates for the numerical approximation for affine optimal control problems subject to semilinear elliptic PDEs. To investigate the error estimates, we focus on local minimizers that satisfy certain local growth conditions. The local growth conditions we consider in this paper appeared recently in the context of solution stability and contain the joint growth of the first and second variation of the objective functional. These growth conditions are especially meaningful for affine control constrained optimal control problems because the first variation can satisfy a local growth, which is not the case for unconstrained problems. The main results of this paper are the achievement of error estimates for the numerical approximations generated by a finite element scheme with piecewise constant controls or a variational discretization scheme. Even though the growth conditions considered are weaker than those appearing in the recent literature on finite element error estimates for affine problems, this paper substantially improves the existing error estimates for both the optimal controls and the states when a Hölder-type growth is assumed.

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arXiv2024

Stability analysis of the Navier-Stokes velocity tracking problem with bang-bang controls

This paper focuses on the stability of solutions for a velocity-tracking problem associated with the two-dimensional Navier-Stokes equations. The considered optimal control problem does not possess any regularizer in the cost, and hence bang-bang solutions can be expected. We investigate perturbations that account for uncertainty in the tracking data and the initial condition of the state, and analyze the convergence rate of solutions when the original problem is regularized by the Tikhonov term. The stability analysis relies on the Hölder subregularity of the optimality mapping, which stems from the necessary conditions of the problem.

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arXiv2024

Analysis of Unregularized Optimal Control Problems Constrained by the Boussinesq System

This paper investigates solution stability properties of unregularized tracking-type optimal control problems constrained by the Boussinesq system. In our model, the controls may appear linearly and distributed in both of the equations that constitute the Boussiniesq system and in the objective functional. We establish, not only the existence of weak solutions, but also unique existence of strong solutions in $L^p$ sense for the Boussinesq system as well as its corresponding linearized and adjoint systems. The optimal control problem is then analyzed by providing the existence of an optimal control, and by establishing first-order necessary and second order sufficient conditions. Then, using assumptions on the joint growth of the first and second variations of the objective functional, we prove the strong metric Hölder subregularity of the optimality mapping, which in turn allows the study of solution stability of the optimal control and states under various linear and nonlinear perturbations. Such perturbations may appear in the Boussinesq system and the objective functional. As an application, we provide a convergence rate for the optimal solutions of the Tikhonov regularized problem as the Tikhonov parameter tends to zero. Furthermore, the obtained stability of the optimal states provides, to the best knowledge of the authors, the first result on the stability of the second-order sufficient condition in affine PDE-constrained optimization under an assumption on the desired profile which is natural for tracking-type objective functionals.

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arXiv2023

New assumptions for stability analysis in elliptic optimal control problems

This paper is dedicated to the stability analysis of the optimal solutions of a control problem associated with a semilinear elliptic equation. The linear differential operator of the equation is neither monotone nor coercive due to the presence of a convection term. The control appears only linearly, or even it can not appear in an explicit form in the objective functional. Under new assumptions, we prove Lipschitz stability of the optimal controls and associated states with respect to perturbations in the equation and the objective functional as well as with respect to the Tikhonov regularization parameter.

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arXiv2022

Solution stability of parabolic optimal control problems with fixed state-distribution of the controls

The paper presents results about strong metric subregularity of the optimality mapping associated with the system of first-order necessary optimality conditions for a problem of optimal control of a semilinear parabolic equation. The control has a predefined spatial distribution and only the magnitude at any time is a subject of choice. The obtained conditions for subregularity imply, in particular, sufficient optimality conditions that extend the known ones. The paper is complementary to a companion one by the same authors, in which a distributed control is considered.

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arXiv2022

Stability in affine optimal control problems constrained by semilinear elliptic partial differential equations

This paper investigates stability properties of affine optimal control problems constrained by semilinear elliptic partial differential equations. This is done by studying the so called metric subregularity of the set-valued mapping associated with the system of first order necessary optimality conditions. Preliminary results concerning the differentiability of the functions involved are established, especially the so-called switching function. Using this ansatz, more general nonlinear perturbations are encompassed, and under weaker assumptions, than the ones previously considered in the literature on control constrained elliptic problems. Finally, the applicability of the results is illustrated with some error estimates for the Tikhonov regularization.

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arXiv2022

On the solution stability of parabolic optimal control problems

The paper investigates stability properties of solutions of optimal control problems for semilinear parabolic partial differential equations. Hölder or Lipschitz dependence of the optimal solution on perturbations are obtained for problems in which the equation and the objective functional are affine with respect to the control. The perturbations may appear in both the equation and in the objective functional and may non-linearly depend on the state and control variables. The main results are based on an extension of recently introduced assumptions on the joint growth of the first and second variation of the objective functional. The stability of the optimal solution is obtained as a consequence of a more general result obtained in the paper -- the proved metric subregularity of the mapping associated with the system of first-order necessary optimality conditions. This property also enables error estimates for approximation methods. Lipschitz estimate for the dependence of the optimal control on the Tikhonov regularization parameter is obtained as a by-product.

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